Found problems: 85335
2008 South africa National Olympiad, 2
Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.
2024 JHMT HS, 7
Compute the sum of all real solutions $\alpha$ (in radians) to the equation
\[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]
2021 Nigerian Senior MO Round 2, 2
$N$ boxes are arranged in a circle and are numbered $1,2,3,.....N$ In a clockwise direction. A ball is assigned a number from${1,2,3,....N}$ and is placed in one of the boxes.A round consist of the following; if the current number on the ball is $n$, the ball is moved $n$ boxes in the clockwise direction and the number on the ball is changed to $n+1$ if $n<N$ and to $1$ if $n=N$. Is it possible to choose $N$, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly $2020$ rounds
PEN H Problems, 43
Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.
II Soros Olympiad 1995 - 96 (Russia), 10.5
Each of the lateral sides of the trapezoid, whose bases are equal to $ a$ and $b$, serves as a side of a regular triangle. One triangle is located entirely outside the trapezoid, and the other has common points with it. Find the distance between the centers of these triangles.
2023 LMT Fall, 10
Aidan and Andrew independently select distinct cells in a $4 $ by $4$ grid, as well as a direction (either up, down, left, or right), both at random. Every second, each of them will travel $1$ cell in their chosen direction. Find the probability that Aidan and Andrew willmeet (be in the same cell at the same time) before either one of them hits an edge of the grid. (If Aidan and Andrew cross paths by switching cells, it doesn’t count as meeting.)
2017 CMIMC Individual Finals, 3
The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points. Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$. What is the length of the segment $\mathcal P$ cuts the $y$-axis?
2015 Peru IMO TST, 11
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
2002 Tuymaada Olympiad, 1
Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon.
Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends?
[i]Proposed by A. Golovanov[/i]
2009 Germany Team Selection Test, 2
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
2018 Israel National Olympiad, 3
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.
1961 Polish MO Finals, 2
Prove that if $ a + b = 1 $, then $$
a^5 + b^5 \geq \frac{1}{16}$$
MBMT Team Rounds, 2020.37
Fuzzy likes isosceles trapezoids. He can choose lengths from $1, 2, \dots, 8$, where he may choose any amount of each length. He takes a multiset of three integers from $1, \dots, 8$. From this multiset, one length will become a base length, one will become a diagonal length, and one will become a leg length. He uses each element as either a diagonal, leg, or base length exactly once. Fuzzy is happy if he can use these lengths to make an isosceles trapezoid such that the undecided base has nonzero rational length. How many multiset choices can he make? (Multisets are unordered)
[i]Proposed by Timothy Qian[/i]
2006 Greece Junior Math Olympiad, 4
If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$
[u]Babis[/u]
[b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]
2024 Turkey Team Selection Test, 4
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.
2025 6th Memorial "Aleksandar Blazhevski-Cane", P2
Let $\triangle ABC$ be a scalene and acute triangle in which the angle at $A$ is second largest, $H$ is the orthocenter, and $k$ is the circumcircle with center $O$. Let the circumcircle of $\triangle AHO$ intersect the sides $AB$ and $AC$ again at $M$ and $N$, respectively, whereas the altitudes $CH$ and $BH$ intersect $k$ again at $K$ and $L$, respectively. Prove that the intersection of $KL$ and the perpendicular bisector of $AH$ is the orthocenter of $\triangle AMN$.
Proposed by [i]Ilija Jovcevski[/i]
1940 Putnam, B2
A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ($r\leq R$) so that the axes intersect at right angles.
i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form
$$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.$$
ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$.
show that
$$S=8[R^2 E - (R^2 - r^2 )K].$$
2012 Dutch IMO TST, 1
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
2013 AIME Problems, 1
Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ $\textsc{am}$, and $7{:}50$ at the former $6{:}00$ $\textsc{pm}$. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ $\textsc{am}$ would have to set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A} + 10\text{B} + \text{C}$.
2003 China Girls Math Olympiad, 1
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that
(1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$
(2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
2019 USA EGMO Team Selection Test, 4
For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that
\[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\]
for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.
2015 European Mathematical Cup, 2
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$
where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?
Solve the problem for:
[list=a][*]$m = n = p = 2,$
[*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list]
[i]Stijn Cambie[/i]
1998 All-Russian Olympiad Regional Round, 10.5
Solve the equation $\{(x + 1)^3\} = x^3$, where $\{z\}$ is the fractional part of the number z, i.e. $\{z\} = z - [z]$.
2023 Durer Math Competition Finals, 2
Timi was born in $1999$. Ever since her birth how many times has it happened that you could write that day’s date using only the digits $0$, $1$ and $2$? For example, $2022.02.21$. is such a date.
2019 Philippine TST, 2
In a triangle $ABC$ with circumcircle $\Gamma$, $M$ is the midpoint of $BC$ and point $D$ lies on segment $MC$. Point $G$ lies on ray $\overrightarrow{BC}$ past $C$ such that $\frac{BC}{DC} = \frac{BG}{GC}$, and let $N$ be the midpoint of $DG$. The points $P$, $S$, and $T$ are defined as follows:
[list = i]
[*] Line $CA$ meets the circumcircle $\Gamma_1$ of triangle $AGD$ again at point $P$.
[*] Line $PM$ meets $\Gamma_1$ again at $S$.
[*] Line $PN$ meets the line through $A$ that is parallel to $BC$ at $Q$. Line $CQ$ meets $\Gamma$ again at $T$.
[/list]
Prove that the points $P$, $S$, $T$, and $C$ are concyclic.